Fear of failure is the main reason one would not chat up an attractive girl. The point of this article is that men think they are going to fail, not because they are not attractive, but because attractive girls must have so many other options.
Therefore, it is somewhat independent of the man's assessment of his own attractiveness. This model is probably oversimplified, however, as self-confidence must play a role in there somewhere.
Carol situation is a case of prisoner's dilemma, which may be presented in a bit shorter way than the article does. Let's use standard game theory terms for the possible actions of one male individual:
'Cooperate' (C) = to publicly approach an attractive girl
'Defect' (D) = to ignore the girl, read the paper instead
So, imagine yourself walking into a cafe and seeing Carol at a table. She has been there for a while, and you have no way of knowing if some other guy has already approached her (or if Carol has a boyfriend). Let's assume for the sake of simplicity that the first man who dares to approach Carol always gets her telephone number (her nice way to reward him for being brave). That automatically means that the second guy will be rejected and feel bad.
The outcomes are (in order of preference):
CD = You dare to speak to Carol, you are the winner, the other guys didn't dare to approach her.
DD = Every man in the room reads the paper, Carol sits alone.
DC = You choose to ignore the girl, then some lucky bastard walks across the room and gets her number. Bad, but you have not lost your face.
CC = The worst outcome. You have approached Carol, but she is already involved with someone else. You walk back to your table, feeling miserable.
You notice that second and third outcome are somewhat identical, there is no gain, but also no real damage. A rational person may well prefer to do nothing and land in the 'neutral' area where 50% of total outcomes lay, rather than to approach Carol and to have 25% chance of success vs. 25% chance of a complete disaster.
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I think it is not the women's attractiveness that acts as a repellent, but uncertainty of her status (has or has not a partner already). The way for Carol so improve her situation is to develop some sort of signalling that says she doesn't mind to be approached (flirting used to do the trick in old times).
This analysis is just wrong. There is no shame or misery in finding out that Carol already has a partner. The really bad outcome is that she's totally alone, but she finds you such a dork, she humiliates you with a zinger that everyone around can hear. Also, women give their phone number to more than one man. They don't have a "first man only" rule.
The reason a lot of guys, especially nerds, won't approach Carol is that they see her as holding higher social status than they do, so it feels extremely uncomfortable. The "popular" (high-status) guys will approach her confidently and playfully. People tend to group with others of approximately the same social status.
To really learn about this stuff, you have to put the math book away and socialize with people.
BTW, the actual policy, as far as I can tell from a couple years experimenting "in the field", is that women's policy for giving out their phone numbers is: they give them to men who ask.
I know, it seems too hard to be true. I had to work up my courage for months to ask for a phone number upon meeting a woman, even after I decided that I would try it since I had nothing to lose. The first time I tried it, the woman gave me her number. The second time I tried it, that one gave me her number, too. And on and on and on. I've only been turned down a few times. And I'm a nervous, skinny, nerdy guy.
I also got a couple bogus numbers. (I count those under "refusals".) It's amazing how most women will avoid humiliating you in public even if they don't want to give you their number.
Once, I asked a lesbian for her phone number, in a bar, right in front of her girlfriend. She gave it to me. It wasn't bogus! She never returned my call, but it's an amazing data point about how women will usually give you their phone number if you ask.
You are forgetting the risk nature of potential suitors.
If the suitors are all risk averse, that is if they perceive that the pain caused by rejection is greater than the pleasure of success, none of them will approach her. On the other hand, if the suitors are risk seeking (or even risk neutral) where pain <= success then there will be those who approach her.
There is a lot to be said regarding their perceived probability of success, and the distribution of perceptions. If unattractive and uneventful suitors are all risk seeking, chances are that they will approach Carol and be promptly rejected.
>Therefore, it is somewhat independent of the man's assessment of his own attractiveness.
The article does mention Guy's evaluation of himself as a candidate. And, intuitively, if Guy considers himself to be extremely attractive, then he should consider p(a) to be accordingly higher.
Keep in mind that it works the other way, too: extreme attractiveness can also be a repellent for women. Especially if the guy has a big ego, is a player, etc. (not many nerds are, but some are). When it comes to relationships, most women I know of prefer loyalty over physical attractiveness, even if they say otherwise.
Of course, there is also the issue of persistence, which cannot be entirely discounted.
Therefore, it is somewhat independent of the man's assessment of his own attractiveness. This model is probably oversimplified, however, as self-confidence must play a role in there somewhere.